Solution
The Ratio of Two Normals
Show the problem again
X and Y are independent standard normal random variables. What is the probability that X/Y < √3? Give a reduced fraction.
Worked solution
The answer is 5/6. The pair (X, Y) is rotationally symmetric, so its direction angle Θ is uniform on [0, 2π). Writing (X, Y) = R(cos Θ, sin Θ) gives X/Y = cot Θ, and as Θ is uniform, X/Y has the standard Cauchy distribution with cumulative distribution function F(c) = 1/2 + arctan(c)/π. Therefore P(X/Y < √3) = 1/2 + arctan(√3)/π = 1/2 + (π/3)/π = 5/6. The ratio of two independent standard normals has no mean at all, yet questions about its distribution reduce to measuring angles.
Source: Classic Gaussian exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.